Abstract
The current paper introduces new prior distributions on the zero-mean multivariate Gaussian model, with the aim of applying them to the classification of covariance matrices populations. These new prior distributions are entirely based on the Riemannian geometry of the multivariate Gaussian model. More precisely, the proposed Riemannian Gaussian distribution has two parameters, the centre of mass \(\bar{Y}\) and the dispersion parameter \(\sigma \). Its density with respect to Riemannian volume is proportional to \(\exp (-d^2(Y; \bar{Y}))\), where \(d^2(Y; \bar{Y})\) is the square of Rao’s Riemannian distance. We derive its maximum likelihood estimators and propose an experiment on the VisTex database for the classification of texture images.
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References
Amari, S., Nagaoka, H.: Methods of Information Geometry. American Mathematical Society, Providence (2000)
Porikli, F., Tuzel, O., Meer, P.: Covariance tracking using model update based means on Riemannian manifolds. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 728–735 (2006)
Kurtek, S., Klassen, E., Ding, Z., Avison, M.J., Srivastava, A.: Parameterization-invariant shape statistics and probabilistic classification of anatomical surfaces. In: Székely, G., Hahn, H.K. (eds.) IPMI 2011. LNCS, vol. 6801, pp. 147–158. Springer, Heidelberg (2011)
Gu, X., Deng, J., Purvis, M.: Improving superpixel-based image segmentation by incorporating color covariance matrix manifolds. In: International Conference on Image Processing (ICIP), pp. 4403–4406 (2014)
Said, S., Bombrun, L., Berthoumieu, Y.: New Riemannian priors on the univariate normal model. Entropy 16(7), 4015–4031 (2014)
Weickert, J., Hagen, H. (eds.): Visualization and Processing of Tensor Fields. Mathematics Visualization. Springer, Heidelberg (2006)
Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982)
Atkinson, C., Mitchell, A.: Rao’s distance measure. Sankhya Ser. A 43, 345–365 (1981)
Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications II. Springer, New York (1988)
Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Americal Mathematical Society, Providence (2001)
Pennec, X.: Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements. J. Math. Imaging Vis. 25(1), 127–154 (2006)
Lenglet, C., Rousson, M., Deriche, R., Faugeras, O.: Statistics on the manifold of multivariate normal distributions. J. Math. Imaging Vis. 25(3), 423–444 (2006)
Mengersen, K., Robert, C., Titterington, M.: Mixtures : Estimation and Applications. Wiley, Chichester (2011)
Saravanan, T.: MIT Vision and Modeling Group. Vision Texture. VisTex : Vision Texture Database. http://vismod.media.mit.edu/pub/vistex
Barachant, A., Bonnet, S., Congedo, M., Jutten, C.: Multiclass brain-computer interface classification by Riemannian geometry. IEEE Trans. Biomed. Eng. 59(4), 920–928 (2012)
Grigorescu, S., Petkov, N., Kruizinga, P.: Comparison of texture features based on Gabor filters. IEEE Trans. Im. Proc. 11(10), 1160–1167 (2002)
Said, S., Bombrun, L., Berthoumieu, Y.: Texture classification using Rao’s distance: an EM algorithm on the Poincaré half plane. In: International Conference on Image Processing (ICIP) (2015)
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Said, S., Bombrun, L., Berthoumieu, Y. (2015). Texture Classification Using Rao’s Distance on the Space of Covariance Matrices. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2015. Lecture Notes in Computer Science(), vol 9389. Springer, Cham. https://doi.org/10.1007/978-3-319-25040-3_40
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DOI: https://doi.org/10.1007/978-3-319-25040-3_40
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